Subgroups — C₂×#21425854 #28741671

Order 15104.

734 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273667	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837357	C₂	1	1	yes	maximal, characteristic
3	#21425854	C₂	1	8	yes	maximal, direct product factor
4	C₂×Dic₅₉⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂³×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11745664	C₄	1	1	yes	characteristic
7	C₂×#14273686	C₄	1	1	yes	characteristic
8	C₂×#14273667	—	2	4	no	—
9	C₂×#14273667	C₂²	1	4	yes	—
10	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
11	#16837357	C₄	1	4	yes	—
12	#16837357	C₂²	1	4	yes	—
13	C₂×#14273637	C₄	1	2	yes	—
14	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
15	Dic₅₉⋊C₁₆	C₂²	1	16	yes	complemented
16	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₅₉	—	2	4	no	—
18	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₅₉	C₂²	1	4	yes	—
20	C₂×#11745664	—	2	4	no	—
21	C₂×#11745664	C₂×C₄	1	4	yes	—
22	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
23	#14273686	C₈	1	4	yes	—
24	#14273686	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₅₉	C₈	1	2	yes	—
26	C₂×C₂²⋊Dic₅₉	C₈	1	2	yes	—
27	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
28	#14273667	—	2	8	no	—
29	C₂×C₅₉⋊C₁₆	—	2	4	no	—
30	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
31	#14273637	C₂×C₄	1	8	yes	—
32	C₈×Dic₅₉	C₂×C₄	1	4	yes	—
33	C₈×Dic₅₉	C₂³	1	4	yes	—
34	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₉	D₄	1	4	yes	—
38	C₂²×C₈×C₅₉	Q₈	1	4	yes	—
39	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₂×C₁₆×C₅₉	—	2	4	no	—
42	C₂×C₁₆×C₅₉	—	2	8	no	—
43	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
44	C₂⁴×C₅₉	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₉	M₁₆	1	4	yes	—
49	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
51	C₂²×Dic₅₉	C₂×C₈	1	4	yes	—
52	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₅₉	C₄⋊C₄	1	8	yes	—
54	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
58	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
61	C₂²⋊Dic₅₉	C₂×C₈	1	8	yes	—
62	C₄⋊Dic₅₉	C₂×C₈	1	8	yes	—
63	C₄×Dic₅₉	C₂×C₈	1	4	yes	—
64	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
65	#11745664	—	2	8	no	—
66	C₂×C₅₉⋊C₈	—	2	4	no	—
67	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
68	C₁₆×C₅₉	—	2	4	no	—
69	C₅₉⋊C₁₆	—	2	4	no	—
70	C₂³×C₅₉	—	2	4	no	—
71	C₂³×C₅₉	C₄⋊C₈	1	4	yes	—
72	C₂³×C₅₉	C₂²×C₈	1	2	yes	—
73	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₄⋊C₈	1	8	yes	—
75	C₂×C₄×C₅₉	C₄⋊C₈	1	4	yes	—
76	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₅₉	C₂×M₁₆	1	4	yes	—
82	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
83	C₂×Dic₅₉	—	2	8	no	—
84	C₂×Dic₅₉	C₂²×C₈	1	8	yes	—
85	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
86	C₈×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
87	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
89	C₅₉⋊C₈	—	2	4	no	—
90	C₂×#2151	—	59	1	no	maximal, Sylow
91	C₂²×C₅₉	—	2	4	no	—
92	C₂²×C₅₉	—	2	8	no	—
93	C₂²×C₅₉	C₂×C₄⋊C₈	1	4	yes	—
94	C₂²×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
95	C₂²×C₅₉	#504	1	4	yes	—
96	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
97	C₄×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
98	C₄×C₅₉	C₂×C₄⋊C₈	1	4	yes	—
99	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
101	Dic₅₉	—	2	8	no	—
102	C₂²×M₃₂	—	59	1	no	—
103	C₂³×C₁₆	D₅₉	1	1	yes	characteristic
104	C₂×#405	—	59	1	no	—
105	#2151	—	59	8	no	—
106	C₂×C₄⋊C₁₆	—	59	4	no	—
107	C₂×C₅₉	—	2	4	no	—
108	C₂×C₅₉	#2151	1	4	yes	complemented
109	C₂×C₅₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₅₉	C₂×#504	1	2	yes	—
111	C₂²×M₁₆	—	59	1	no	—
112	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
113	C₂×#117	—	59	1	no	—
114	C₂×M₃₂	—	59	4	no	—
115	C₂×M₃₂	—	118	4	no	—
116	C₂²×C₁₆	—	2	4	no	—
117	C₂²×C₁₆	—	59	2	no	—
118	C₂²×C₁₆	C₂×D₅₉	1	4	yes	—
119	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
120	#405	—	59	4	no	—
121	#405	—	59	4	no	—
122	C₂×#97	—	59	2	no	—
123	C₂×C₄×C₈	—	59	2	no	—
124	C₄⋊C₁₆	—	59	16	no	—
125	C₅₉	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
127	C₂×M₁₆	—	59	4	no	—
128	C₂×M₁₆	—	118	4	no	—
129	C₂²×C₈	—	59	2	no	—
130	C₂²×C₈	Dic₁₁₈	1	4	yes	—
131	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₅₉	1	2	yes	—
133	C₂²×C₈	C₂²⋊D₅₉	1	4	yes	—
134	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
135	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
136	#117	—	59	4	no	—
137	#117	—	59	4	no	—
138	C₂×C₄⋊C₄	—	59	2	no	—
139	C₂×C₂²⋊C₄	—	59	2	no	—
140	C₂×C₄²	—	59	2	no	—
141	M₃₂	—	118	8	no	—
142	C₂×C₁₆	—	2	4	no	—
143	C₂×C₁₆	—	2	8	no	—
144	C₂×C₁₆	—	59	4	no	—
145	C₂×C₁₆	—	118	4	no	—
146	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
147	#97	—	59	8	no	—
148	C₄×C₈	—	59	4	no	—
149	C₄×C₈	—	59	4	no	—
150	C₂⁴	C₈×D₅₉	1	1	yes	characteristic
151	C₂²×C₄	—	59	4	no	—
152	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
153	C₂²×C₄	C₈×D₅₉	1	2	yes	—
154	C₂²×C₄	#11745659	1	4	yes	—
155	C₂²×C₄	Dic₅₉⋊C₄	1	4	yes	—
156	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
158	M₁₆	—	118	8	no	—
159	C₂×C₈	—	59	4	no	—
160	C₂×C₈	—	118	4	no	—
161	C₂×C₈	Dic₅₉⋊C₄	1	4	yes	—
162	C₂×C₈	Dic₅₉⋊C₄	1	8	yes	—
163	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
164	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
165	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
166	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
167	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
168	C₂×C₈	C₂×C₂²⋊D₅₉	1	4	yes	—
169	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
170	C₄⋊C₄	—	59	8	no	—
171	C₂²⋊C₄	—	59	8	no	—
172	C₄²	—	59	4	no	—
173	C₄²	—	59	4	no	—
174	C₁₆	—	2	4	no	—
175	C₁₆	—	118	4	no	—
176	C₂³	—	2	4	no	—
177	C₂³	Dic₅₉⋊C₈	1	4	yes	—
178	C₂³	C₂×C₈×D₅₉	1	2	yes	—
179	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
180	C₂×C₄	—	59	8	no	—
181	C₂×C₄	—	118	8	no	—
182	C₂×C₄	Dic₅₉⋊C₈	1	8	yes	—
183	C₂×C₄	Dic₅₉⋊C₈	1	4	yes	—
184	C₂×C₄	C₂×Dic₅₉⋊C₄	1	4	yes	—
185	C₂×C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
186	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
188	C₂×C₄	C₂×#11745659	1	2	yes	—
189	C₂×C₄	C₂×#11745659	1	4	yes	—
190	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
191	C₈	—	118	4	no	—
192	C₈	C₂×Dic₅₉⋊C₄	1	4	yes	—
193	C₈	C₂×Dic₅₉⋊C₄	1	2	yes	—
194	C₈	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
195	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
196	C₂²	—	2	4	no	—
197	C₂²	—	2	8	no	—
198	C₂²	C₂×Dic₅₉⋊C₈	1	4	yes	—
199	C₂²	C₂×Dic₅₉⋊C₈	1	2	yes	—
200	C₂²	#16837818	1	4	yes	—
201	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
202	C₄	—	118	8	no	—
203	C₄	C₂×Dic₅₉⋊C₈	1	2	yes	—
204	C₄	C₂×Dic₅₉⋊C₈	1	4	yes	—
205	C₄	C₂²×Dic₅₉⋊C₄	1	1	yes	characteristic
206	C₄	C₂²×#11745659	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#21425854	1	4	yes	direct product factor
209	C₂	C₂²×Dic₅₉⋊C₈	1	1	yes	characteristic
210	C₂	C₂×#16837818	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂³×C₁₆×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₈
prime radical	C₂⁴×C₅₉	C₂×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂³×C₁₆	D₅₉
hypocenter	C₅₉	C₂×#2151
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #21425854	C₂	direct (N × H)
4. C₂×Dic₅₉⋊C₁₆	C₂	semidirect
15. Dic₅₉⋊C₁₆	C₂²	semidirect
108. C₂×C₅₉	#2151	semidirect
125. C₅₉	C₂×#2151	semidirect
208. C₂	#21425854	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14273667	2
C₂×#16837357	2
#21425854	2
C₂×Dic₅₉⋊C₁₆	2
C₂³×C₁₆×C₅₉	2
C₂×#2151	59

Subgroups — C2×#21425854 #28741671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21425854 #28741671